Let us now consider how the phase-shifts in Eq. (1306) can be
evaluated. Consider a spherically symmetric potential which
vanishes for , where is termed the range of the potential.
In the region , the wavefunction
satisfies the free-space Schrödinger equation (1285). The
most general solution which is consistent with no incoming spherical-waves is

(1309)

where

(1310)

Note that functions are allowed to appear in the above
expression, because its region of validity does not include the origin
(where ). The logarithmic derivative of the th
radial wavefunction,
, just outside the range of the potential is given by

(1311)

where denotes , etc. The above equation
can be inverted to give

(1312)

Thus, the problem of determining the phase-shift is equivalent
to that of obtaining .

The most general solution to Schrödinger's equation inside
the range of the potential () which does not depend on the
azimuthal angle is

(1313)

where

(1314)

and

(1315)

The boundary condition

(1316)

ensures that the radial wavefunction is well-behaved at the
origin.
We can launch a well-behaved solution of the above equation from
, integrate out to , and form the logarithmic derivative

(1317)

Since and its first derivatives are necessarily continuous for
physically acceptible wavefunctions, it follows that